1. In a series R L circuit, we get this equation for the current i : i(t) = E/R[1-e^(-Rt/L)] where R: Resistance, L: inductance, E: emf

This current is the response to a step input of voltage. No other input will yield this response. This IS the transient response. The steady-state response is i = E/R. In this case V = step input voltage V0 U(t).

The basic differential equation is V = Ri + L di/dt, based on the simple fact that for an inductorr V = L di/dt and for a resistor V = iR.

If a sinusoidal voltage V = V0 sin(wt) is applied at t=0 there is a transient as well as a steady-state response. The above diffrerential equation allows solving for both.